The Zernike polynomials are an inﬁnite set of orthogonal polynomials over the unit
disk, which are rotationally invariant. They are frequently utilized in optics, opthal-
mology, and image recognition, among many other applications, to describe spherical aberrations and image features.
Discretizing the continuous polynomials, however,
introduces errors that corrupt the orthogonality. Minimizing these errors requires numerical considerations which have not been addressed.
This work examines the
orthonormal polynomials visually with the Gram matrix and computationally with the rank and condition number. The convergence of the Fourier-Zernike coeﬃcients and the Fourier-Zernike series are also examined using various measures of error.
The orthogonality and convergence are studied over six grid types and resolutions, polynomial truncation order, and function smoothness.
The analysis concludes with
design criteria for computing an accurate analysis with the discrete Zernike polynomials.